Article

Selecta Mathematica

, 13:369

First online:

Gelfand transforms and Crofton formulas

  • J. C. Álvarez PaivaAffiliated withLaboratoire Paul Painlevé, Université des Sciences et Technologies de Lille Email author 
  • , E. FernandesAffiliated withMULTITEL Research Centre, Parc Initialis

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Abstract.

The term integral geometry has come to describe two different fields of research: one, geometrical, based on the works of Blaschke, Chern, and Santaló, and another, analytical, based on the works of Radon, John, Helgason, and Gelfand. In this paper we bridge the gap by showing that classical integral-geometric formulas such as those of Crofton, Cauchy, and Chern can be easily and systematically obtained through the study of Radon-type transforms on double fibrations. The methods also allow us to extend these formulas to non-homogeneous settings where group-theoretic techniques are no longer useful. To illustrate this point, we construct all Finsler metrics on projective space such that hyperplanes are area-minimizing and extend the theory of Crofton densities developed by Busemann, Pogorelov, Gelfand, and Smirnov.

Pensar es olvidar diferencias. . .

Jorge Luis Borges

Mathematics Subject Classification (2000).

Primary 53C65 Secondary 53C60

Keywords.

Integral geometry Crofton formulas double fibrations Gelfand transforms Finsler manifold Holmes–Thompson area